approximately in the case of Mars.

It is evident from the parameters used by Copernicus and Ptolemy (given in Table 5.1 in terms of the length |CO|, which is fixed at 10 000) that, in a strictly geometrical sense, there is not much to choose between the two schemes. However, the Copernican scheme, as a scientific theory, has a great advantage over its Ptolemaic equivalent (though not one that was particularly influential in the sixteenth century). In the geostationary scheme shown inFigure 5.6, one must make the ad hoc assumption that ES and C"P are parallel in order to link the planet's motion correctly to that of the Sun, and this must be done for all of the planets. On the other hand, in the heliocentric theory, all these assumptions are replaced by a single proposition - that the Earth orbits the Sun.

Another powerful way of making the same point is illustrated in Table 5.2. For each of the three superior planets, the table shows the zodiacal period Tc, the synodic period Tp, and the angular speeds to which they correspond, &>c and ¬ęp, respectively. In each case, if we add &>c and together, we get a rate of 1 revolution per year. Why? According to Ptolemy, &>c + &>p is the rate at which the planet revolves around its epicycle relative to a fixed reference line (the planet rotates around the centre of its epicycle at the rate &>p, but the centre of the epicycle is itself rotating at the rate &>c). Since &>c + &>p = 1, Ptolemy makes

The true orbit is an ellipse with eccentricity e, for example. A simple argument using complex numbers can be used to show that both Copernicus' and Ptolemy's theories give planetary positions accurate to first order in e (see, for example, Hoyle (1974)).

Table 5.2. The periods of the superior planets.

Table 5.2. The periods of the superior planets.

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